Fractal bodies invisible in 2 and 3 directions
Alexander Plakhov
∗Vera Roshchina
†Abstract
We study the problem of invisibility for bodies with mirror surface in the frame-work of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal direc-tions. The work continues the previous study in [1, 10], where two-dimensional bodies invisible in one direction and three-dimensional bodies invisible in one and two orthogonal directions were constructed.
Mathematics subject classifications: 37D50, 49Q10
Key words and phrases: Billiards, invisible bodies, shape optimization, geomet-rical optics, problems of minimal resistance.
1
Introduction
Invisibility has fascinated people’s imagination since ancient times: the idea is exploited in folklore, fiction and movies. This “magical” concept is, however, rapidly migrating into the scientific domain. The cutting edge of the modern developments is the design of metamaterials with special refractive properties, which can ultimately lead to the creation of a real invisible cloak, not mentioning other important applications. For an overview of the recent works in this field we refer the reader to our recent article [10]. The aforemen-tioned developments deal with the wave nature of light, and metamaterials are engineered at the nanoscale level. The effects specific to geometrical optics, however, also remain im-portant in modern technology, mostly in the cases when the objects are large enough for geometrical optics to dominate the wave effects. Examples include fiber optics, design of lenses (e.g. for photography or DVD readers), and many others.
In this article we are concerned with invisibility in billiards. We consider bodies with a perfectly mirror surface in a beam of light, or, equivalently, in a flow of non-interacting
∗Department of Mathematics, University of Aveiro, Aveiro 3810-193, Portugal †CIMA-UE, University of ´Evora, Portugal
billiard particles. Invisibility in a direction v (where v is a unit vector) means that any incident light ray which initially moves along a straight line in this direction, after several reflections from the body’s surface will eventually move along the same straight line. Invisibility in a set of directions means that the above is true for any direction from this set. This problem is closely related to the problem of minimal resistance going back to Newton [7]. The latter consists of finding a body, from a given class of bodies, that experiences the smallest possible force of flow pressure, or resistance force. Since 1990s, many interesting results on this problem have been obtained by various authors (see, e.g., [2, 3, 4, 5, 6, 8, 9]). Bodies of zero resistance in one and two directions are described in [1] and [10] respectively. In both cases it is possible to construct an invisible body by arranging several such bodies together in a specific way. In [10] it was shown that there exist bodies invisible in two mutually orthogonal directions in the three-dimensional setting and that bodies invisible in all directions do not exist.
In this work we continue the study of invisibility and construct bodies invisible in any two directions in two-dimensional space and in three orthogonal directions in three-dimensional setting. Each body in the construction is a disjoint union of infinitely many pieces of size going to zero, where each piece is a domain with piecewise smooth boundary. By slightly abusing the language, such a union will be called a fractal body, or a solid fractal body. In a preliminary construction (Section 3.1) and in a limiting case of the basic construction (Section 4) some pieces comprising the body are smooth curves (in the 2D case) or pieces of surfaces (in the 3D case). The corresponding body will be called a thin fractal body.
The article is organized as follows: we first remind some definitions and briefly revisit earlier results in Section 2. In Sections 3 and 4 we explain the construction of bodies invisible in two and three directions respectively. Section 5 contains some final remarks and a brief discussion of open problems.
2
Bodies of zero resistance and invisible bodies
Consider a parallel flow of point particles in Rn moving with unit velocity v ∈ Sn−1
towards a body B at rest. The flow is so rarefied that the particles do not mutually interact. Particles reflect elastically when colliding with the body surface and move freely between consecutive collisions.
We deal with bounded (not necessarily connected) bodies composed of a (possibly infinite) number of piecewise smooth fragments. When a particle moving along a straight line and with a constant velocity hits the boundary of B, it is reflected from the latter without loss of speed, and keeps moving along the new linear trajectory until the next collision. All reflections are specular: the angle of incidence just before the collision is equal to the angle of reflection just after the collision (see Fig. 1). In general, it is possible that the particle never leaves the body and keeps bouncing off its sides infinitely; however,
ξ
ξ+
v+
v
B
Figure 1: The broken line through the points ξ and ξ+ is a billiard trajectory in the
complement of B.
we only consider such bodies and velocities of incidence v for which almost every particle makes a finite number of reflections. Also note that in some cases the particle may hit a singular point of the boundary. In this case the further movement of the particle is not defined. We consider such bodies and velocities v that the set of particles with the incident velocity v hitting singular points has zero measure.
In view of this description, for almost any ξ ∈ Rn, the particle that initially moves
freely according to x(t) = ξ + vt, after a finite number of reflections from B moves freely again according to x(t) = ξ++ v+t, where ξ+ = ξ+
B,v(ξ) and v+ = v +
B,v(ξ) are measurable
functions defined almost everywhere. Definition 1. Let a body B ⊂ Rn.
(i) We say that B has zero resistance in the direction v, if vB,v+ (ξ) = v for all ξ in the domain of v+B,v (see Fig. 2 (a)).
(ii) We say that the body B is invisible in the direction v, if it has zero resistance in this direction and, additionally, ξB+(ξ, v) − ξ is always parallel to v (see Fig. 2 (b)). (iii) Let A ⊂ Sn−1. The body B is said to be invisible/have zero resistance in the set of
directions A, if it is invisible/has zero resistance in any direction v ∈ A.
Observe that invisibility is a symmetric notion, i.e. if a body is invisible in a direction v, it is also invisible in −v. This follows directly from the fact that billiard dynamics is time-reversible.
Examples of bodies invisible in one direction constructed with the use of thin mirrors can be found on the Wikipedia page on invisibility [11], and one of them is reproduced
v+= v v (a) v ξ ξ+ (b)
Figure 2: A typical billiard path in the case of a body (a) having zero resistance in the direction v; (b) invisible in the direction v. The body is not shown in both cases.
(a) (b)
Figure 3: Bodies invisible in one direction: (a) using two parabolic mirrors and a thin flat mirror; (b) using flat mirrors.
in Fig. 3 (a). The German Wikipedia page [12] has got more interesting examples and videos of prototypes designed by Karl Bednarik (one of them that uses only flat mirrors is plotted in Fig. 3 (b)).
A solid body (i.e. without use of any thin mirrors) of zero resistance in one direction was constructed in [1] (see Fig. 4). It was also shown in the same work that there exist connected (and even homeomorphic to the ball) bodies of zero resistance in one direction. An invisible body is obtained by using two such bodies consecutively (see [1] for details). In [10] a body of zero resistance in two directions in three-dimensional setting was described. The body is sketched in Fig. 5: it employs 8 fragments of congruent parabolic cylinders with two mutually orthogonal focal lines (each line corresponds to 4 fragments). An invisible body is constructed by putting 4 such bodies together, as shown in Fig. 5 (b). For more details we refer the reader to [10].
(a) (b)
Figure 4: Solid body of zero resistance in one direction: (a) two-dimensional construction; (b) three-dimensional version.
(a) (b)
Figure 5: Solid body of zero resistance in two directions.
3
Body invisible in two directions
3.1
A thin fractal body invisible in two orthogonal directions
We start with a two-dimensional body invisible in two orthogonal directions. For the clarity of exposition, we assume that the directions of invisibility are parallel to the x-and y-axes.
We construct our body inside the square
S = [−1, 1] × [−1, 1] = {(x, y) : |x| ≤ 1, |y| ≤ 1}.
Consider a thin parabolic mirror (i.e. having zero thickness) p1 given by the equation
p1 =(x, y) : y =
1 2x
2+ 1
Observe that (−1, 1) ∈ p1, (1, 1) ∈ p1, and the focus of the parabola y = 12x2 + 12 is
located at (0, 1). The axis is then given by the equation x = 0. Also note that the mirror is located above the graph of the function y = |x|.
A particle moving “downwards”, i.e., along the direction (0, −1), would be reflected towards the focal point due to the reflective property of parabola (see Fig. 6 (a)).
(a) (b)
Figure 6: Fractal body invisible in two directions: (a) constructing the first parabolic mirror; (b) adding a similar confocal mirror.
We add one more parabolic mirror p2 to our construction. This mirror is similar to
the previous one, only two times smaller, while the foci of the corresponding parabolas coincide at (0, 1). We let p2 = {(x, y) : y = x2+ 3 4, |x| ≤ 1 2}.
Now all particles that go in the downward direction and pass the line segment [−1, −1
2] × {1}) (as well as [ 1
2, 1] × {1})), are first reflected towards the focal point, move
towards p2, and after the collision with p2 are redirected downwards (because of the
afore-mentioned property of parabola and coinciding foci of the two parabolas). If we remove a piece that is directly behind p2 from p1, the resulting body
p′ 1 = {(x, y) : y = 1 2x 2+ 1 2, 1 2 ≤ |x| ≤ 1},
is not obstructing the further movement of the particles, and they leave the body with the same velocity as they had before entering the body (see Fig. 6 (b)).
If we repeat this construction process infinitely, adding figures similar to p1and cutting
out the middle sections of the relevant parabolas, we obtain a sequence {p′ k} ∞ 1 of parabolic mirror segments: p′ k =(x, y) : y = 2 k−2x2+ 1 − 2−k , 2−k ≤ |x| ≤ 2−k+1 .
This mirror sequence is plotted in Fig. 7 (a). The union of these segments is denoted by P = ∪∞
k=1p ′ k.
Take the sequence of parabolic mirror pieces qk symmetric to p ′
k with respect to the
x-axis,
qk =(x, y) : y = −2k−2x2− 1 + 2 −k
, 2−k≤ |x| ≤ 2−k+1 and let Q = ∪∞
k=1qk; the union P ∪ Q is a fractal body invisible in the direction (0, ±1)
(see Fig. 7 (b)). Indeed, since the lower part Q of our body is symmetric to the upper
(a) (b)
Figure 7: Fractal body invisible in two directions: (a) the basic fractal construction; (b) body invisible in the vertical direction.
part P , it is redirecting the particles back to their original trajectories. Observe that the area
G = {(x, y) : |y| ≤ 1 2x 2 +1 2, 1 2 ≤ |x| ≤ 1}
(greyed in Fig. 7 (b)) is completely “shaded” from the particles moving parallel to y-axis. We can hence use this area to make our body invisible in the second direction. We simply add one more construction identical to the original one, but rotated by π
2. It is not difficult
Also observe that the four grey blocks in the corners of the square (see Fig. 8), are never accessed by particles moving in the directions of invisibility. We can hence include these areas into our body.
Figure 8: A thin fractal body invisible in two orthogonal directions Thus, we have proved the following
Theorem 1. There exists a thin fractal body invisible in two perpendicular directions.
3.2
A body invisible in two arbitrary directions
In this section we go further and construct a two-dimensional fractal body (without thin parts) invisible in any two directions.
This time we construct our body inside a rhombus ABCD with sides parallel to the directions of invisibility. Assume that we are given two non-parallel directions, not necessarily perpendicular. We rotate the coordinate system in a way that one of the directions is vertical. Our to-be-invisible rhombus has got two sides parallel to the y-axis, and the other two parallel to the second direction of invisibility. The center of the rhombus coincides with the origin.
This time our construction requires some preparatory work. Denote by −c the abscissa of A and select two infinite sequences of positive values a0, a1, a2, . . . and c0, c1, c2, . . .
recursively according to the following rules.
Let c0 = c and a0 = +∞, and choose arbitrary c1 satisfying 0 < c1 < c0. On the ith
step of the procedure, i = 1, 2, . . . select arbitrary ai satisfying the inequalities
x y b b b b b b b b b b b b b b b b b b b b A B C D −c0 −c1 −c2 a1 a2 −c′ 1 −c′ 2 a′1 a′ 2 q0 q′ 0 p0 p′ 0 q1 q′ 1 p1 p′ 1
Figure 9: Preliminary construction: A body inside a rhombus invisible in one direction. and put
ci+1 =
(ci+ ai)2
ci−1+ ai
− ai. (2)
Using (1) and (2), one easily gets that ci > 0 and derives by induction that ci+1 < ci.
Now we are ready to draw the parabolas of our construction. In the descrip-tion below, the points on the side AB are marked by the values of their abscissas: −c0, −c1, −c2, . . . , a1, a2, . . . (see Fig. 9).
For any i = 0, 1, 2, . . . consider the parabola through −ci with vertical axis and with
focus at ai+1. Denote by pi the arc of this parabola bounded by the points with abscissas
−ci and −ci+1. Then for any i = 1, 2, . . . consider the parabola through −ci with vertical
abscissas −ci and −ci+1 is denoted by qi. Also denote by q0 the part of the diagonal
AC between A and the point with abscissa −c1. Each arc pi is situated above qi; more
precisely, both pi and qi are the graphs of functions pi(−t) and qi(−t), t ∈ [ci+1, ci] with
pi(−t) > qi(−t) for any t ∈ (ci+1, ci]. Denote by Pi, i = 0, 1, 2, . . . the set bounded by
the arcs pi and qi from above and below and by a segment of the vertical line xi = −ci+1
on the right.
Similarly, denote by −c′ iand a
′
i, respectively, the points on the side CD whose abscissas
are −ci and ai (the same values as above). Denote by p ′ i (q
′
i) the arc of parabola through
−c′
i with vertical axis and with focus at a ′ i+1 (a
′
i) bounded by the vertical lines x = −ci and
x = −ci+1. Introduce the sets P ′
i bounded by p ′ i, q
′
i and the segment of line x = −ci+1.
Denote by Hi (i = 1, 2, . . .) the homothety with the center ai and ratio ri = (ci +
ai)/(ci−1 + ai).
Proposition 1. The arcs pi−1 and qi are homothetic under Hi.
Proof. Note that the parabolas containing pi−1 and qi have the same focus at ai; therefore
they are homothetic with the center at this point. Using that the points −ci−1 and −ci
(which are the left endpoints of the parabolas pi−1 and qi) are homothetic, one readily
concludes that the ratio equals ri, and therefore, the homothety is Hi. Further, the right
endpoint of pi−1 has abscissa −ci; using (2), one verifies that its image under Hi has the
abscissa −ci+1, and therefore, coincides with the right endpoint of qi. Thus, the arcs pi−1
and qi belong to homothetic parabolas, and their right and left endpoints are homothetic;
therefore they are also homothetic under Hi.
Introduce the fractal body (greyed in Figure 9) AL = (∪ ∞ i=0Pi) ∪ (∪ ∞ i=0P ′ i).
Now we are in a position to prove the following Proposition. Proposition 2. The body AL is invisible in the vertical direction.
Proof. Consider a particle falling vertically downward with the velocity (0, −1) along a line with abscissa x. We assume that x ∈ [−c, 0]; otherwise the particle does not hit the body and there is nothing to prove. If x = −ci (i = 0, 1, 2, . . .), the particle hits the body
at a singular point and the motion is not defined since then. Otherwise, x belongs to an interval (ci−1, ci), i = 1, 2, . . ..
The particle is first reflected by pi−1 and then moves to the focus ai; then it makes
the second reflection from qi and moves along a vertical line. Since the line through the
two reflection points contains the focus, these two points (and therefore the vertical lines through these points) are homothetic under Hi.
The particle then makes the third and fourth reflections from q′ i and p
′
i−1, and finally,
A′ B′ C′ D′ A B C D (a) A B C D (b)
Figure 10: (a) A centrally symmetric body invisible in one direction and the shaded regions. (b) A body invisible in two directions.
lines through the third and fourth reflection points are homothetic under the inverse homothety H−1
i , and therefore, the vertical lines containing the initial and final parts of
the trajectory coincide.
By adding the set AR symmetric to AL with respect to the center of the rhombus,
one gets the body A = AL∪ AR, which is also invisible in the vertical direction (0, −1)
(Fig. 10 (a)).
Remark 1. The arcs pi, qi, p ′ i, q
′
i, i = 0, 1, 2, . . . are graphs of functions; denote these
functions by pi(−x), qi(−x), p’i(−x), q’i(−x), x ∈ [ci+1, ci]. Each function takes values
in [c1, c]. Then the sets Pi, P ′
i can be represented as
Pi = {(x, y) : −ci ≤ x ≤ −ci+1, qi(−x) ≤ y ≤ pi(−x)},
Pi′ = {(x, y) : −ci ≤ x ≤ −ci+1, p’i(−x) ≤ y ≤ q’i(−x)}.
In particular, if the directions of invisibility are orthogonal, we have p’i(x) = −pi(x),
q’i(x) = −qi(x), and the body A is analytically described as
A = ∪∞
This representation will be used in the next section. Observe that the trapezoids AA′
D′
D and BB′
C′
C bounded by the lines x = ±c0,
x = ±c1 and by the diagonals of the rhombus are completely ”shaded” from the particles
falling in the vertical direction. We can therefore use this area to make our body invisible in the second direction.
Namely, the body B symmetric to A with respect to the line BD (or AC) is contained in the union of the trapezoids and is invisible in the direction −→AB. The union A ∪ B is then invisible in both directions (Fig. 10 (b)).
We have proved the following result. Observe that it includes Theorem 1 as a particular case.
Theorem 2. For any two directions v1 and v2 ∈ S1, there exists a solid fractal body
invisible in these directions.
Remark 2. In the limiting case ci = 2 −i
c, ai = 0 (i ≥ 1) one gets a thin fractal. In
particular, if the two directions are orthogonal, we have the construction described in the previous Subsection 3.1.
4
A body invisible in 3 directions
Using the two-dimensional construction described in the previous section, we can now describe a three-dimensional body invisible in 3 orthogonal directions.
In the 3D case the construction is more complicated and intuition is less reliable; therefore we provide here a more detailed argument than in the previous section. Some of the accompanying figures, for better visibility, correspond to the limiting case of thin fractal body.
Let us first introduce some notation. We consider Euclidean space R3 with orthogonal
coordinates x, y, z and the cube Q = [−c, c]3 centered at the origin O = (0, 0, 0). The
pyramids with the vertex at O and with the bases z = ±c, |x| ≤ c, |y| ≤ c (the upper and lower sides of the cube Q) are denoted by Π±
z. In other words,
Π±
z = {(x, y, z) ∈ Q : ±z ≥ max{|x|, |y|} }.
Further, Πz = Π+z ∪ Π −
z. The pyramids Πx and Πy are defined in the same way. Notice
that the interiors of Πx, Πy, and Πz ara mutually disjoint.
Let 0 < c1 < c. For each εx ∈ {−1, 1} and εz ∈ {−1, 1} we define the gallery
G(εx, εz) = εx[c1, c] × [−c, c] × εz[c1, c]
(where by definition ε[a, b] = [εa, εb], if ε > 0
[εb, −εa], if ε < 0 ; it is a horizontal parallelepiped adjacent to an edge of Q parallel to the y-axis. The union of 4 such parallelepipeds is
The sets Gx and Gz are defined similarly.
We are going to define three bodies Bx, By, Bz, invisible in the directions x, y, z,
respectively, and then take their union.
First we take the two-dimensional body A = Ayz in the yz-plane, as described in the
previous subsection 3.2. It corresponds to 2 orthogonal directions (parallel to the y- and z-axes), and therefore is inscribed in the square [−c, c]2. (Notice that the body A shown
in Fig. 10 (a) corresponds to 2 non-orthogonal direction and therefore is inscribed in a rhombus.) The body Ayz invisible in the z-direction is determined by the parameters
c0 = c, c1, c2, . . . ; a1, a2, . . . satisfying (1) and (2).
Take the direct product ˜
Ayz = ([−c, −c1] ∪ [c1, c]) × Ayz;
the resulting three-dimensional body ˜Ayz is also invisible in the z-direction. Notice that
it is contained in the union of y-galleries, ˜
Ayz ⊂ Gy. (3)
The body ˜Ayz is shown inFig. 11 (a); for the sake of better visualization the limiting case
of ”thin fractal” with ai = 0 and ci = 2
−ic (i = 1, 2, . . .) is chosen here.
Let Byz = ˜Ayz∩ Πz (Fig. 11 (b)). Then we analogously define the body Bxz and take
Bz = Byz∪ Bxz.
The bodies Bx and By are defined in a similar way, and finally,
B = Bx∪ By∪ Bz.
(a) (b)
Remark 3. In terms of the functions pi and qi introduced in Remark 1, the set ˜Ayz can
be written as ˜ Ayz = ∪
∞
i=0{(x, y, z) : ci+1 ≤ |y| ≤ ci, qi(|y|) ≤ |z| ≤ pi(|y|), c1 ≤ |x| ≤ c}.
Taking the intersection of ˜Ayz with Πz = {(x, y, z) : |z| ≥ |x|, |z| ≥ |y|} and using that
qi(|y|) ≥ |y|, one gets
Byz = ∪ ∞
i=0{(x, y, z) : ci+1 ≤ |y| ≤ ci, qi(|y|) ≤ |z| ≤ pi(|y|), c1 ≤ |x| ≤ |z|},
and therefore,
Byz⊂ ∪ ∞
i=0{(x, y, z) : ci+1 ≤ |y| ≤ ci, |x| ≤ pi(|y|)}. (4)
Similar relations are true for the other sets Byx, Bxz, etc; for example,
Bxz = ∪ ∞
i=0{(x, y, z) : ci+1 ≤ |x| ≤ ci, qi(|x|) ≤ |z| ≤ pi(|x|), c1 ≤ |y| ≤ |z|≤ c}, (5)
Byx ⊂ ∪ ∞
i=0{(x, y, z) : ci+1 ≤ |y| ≤ ci, qi(|y|) ≤ |x| ≤ pi(|y|)}. (6)
Now we are prepared for the proof of the following theorem.
Theorem 3. There exists a solid fractal body invisible in 3 mutually orthogonal directions. Proof. We are going to show that B is invisible in the directions parallel to the coordinate axes.
It suffices to prove that B is invisible in the z-direction; this will imply invisibility in the x- and y-directions, due to symmetry of the construction under the exchange of variables x, y and z. We will actually show that the body Bz is invisible for the incident
flow in the z-direction and the bodies Bx and By are shadowed from this flow by Bz.
Consider an incident particle with the velocity (0, 0, −1). Our goal is to prove that it makes 4 reflections from Bz (and no reflections from Bx and By) and moves freely
afterwards; moreover, the initial and final parts of its trajectory belong to the same straight line.
If the orthogonal projection of the coordinate of the incident particle on the xy-plane lies inside the square [−c1, c1]2 or outside the square [−c, c]2 (see Fig. 12), the particle
does not hit the body and there is nothing to prove. It remains therefore to consider the case where the projection is contained in the set [−c, c]2 \ [−c
1, c1]2.
Due to symmetry of the construction, it suffices to consider the cases where the pro-jection belongs to (i) the triangle c1 < x < y < c and (ii) the rectangle c1 < x < c,
0 < y < c1 (they are shown grey in Fig. 12).
Consider the case (i). Take an auxiliary trajectory corresponding to the particle with the same initial data making reflections from ˜Ayz. At the first point of reflection one has
x y c c1 c2 c −c −c1 c1
Figure 12: The frame of the construction in the xy-projection.
z = p0(y), at the second point, z = q1(y), and the third and fourth reflection points are
symmetric to the first two ones with respect to the xy-plane. After the fourth reflection the particle moves freely. We know that ˜Ayz is invisible in the z-direction; that is, the
initial and final parts of the auxiliary trajectory belong to a straight line. It remains to show that the points of impact actually belong to Byz and the auxiliary trajectory does
not intersect the rest of the body B \Byz; this will imply that it is a true billiard trajectory
in the complement of B.
The parts of the auxiliary trajectory in Q before the first reflection, between the first and the second reflection, and between the second and the third reflection will be referred to as 01-segment, 12-segment, and 23-segment, respectively. At the first reflection point (x(1), y(1), z(1)) one has
z(1) = p0(y(1)) > y(1) > x(1),
therefore this point belongs to the interior of Πz, and so, (x(1), y(1), z(1)) ∈ Byz. At the
second reflection point (x(2), y(2), z(2)) one has x(2) = x(1), 0 < y(2)< y(1), and z(2) > z(1),
therefore this point also belongs to the interior of Πz, and so, (x(2), y(2), z(2)) ∈ Byz.
Since Πz is convex, one concludes that the segment with endpoints (x(1), y(1), z(1)) and
(x(2), y(2), z(2)) also belongs to the interior of Π z.
Using (5) and the inequality c1 < x(1) < c, one concludes that the intersection of Bxz
with the plane x = x(1) belongs to the set {c
1 ≤ y ≤ c, z ≤ p0(x(1))}. On the other hand,
p0(x(1)) < p0(y(1)) and the 01- and 12-segments belong to the set {z ≥ p0(y(1))}, and the
23-segment belongs to the set {y < c1}. This implies that the first three segments of
the trajectory do not intersect with Bxz, and due to the symmetry of both Bxz and the
trajectory with respect to the xy-plane, this is true for the whole trajectory.
Further, the 01- and 12-segments belong to the interior of Πz, and therefore, do not
intersect with Bx and By. Due to the symmetry with respect to the xy-plane, this is also
true for the last two segments of the auxiliary trajectory. It remains therefore to prove that the 23-segment does not intersect with the bodies Bx = Byx∪Bzxand By = Bzy∪Bxy.
The sets Bzx, Bzy and Bxy belong to the galleries Gz and Gx, and therefore cannot
At the second reflection point (x(2), y(2), z(2)) one has z(2) = q
1(y(2)) and z(2) > x(2).
At each point of the 23-segment one has x = x(2), y = y(2), and therefore, x < q
1(y) and
c2 < y < c1. By (6), no such point belongs to Byx.
Consider now the case (ii). Take i ≥ 1 such that ci+1 < y < ci. (The limiting case
y = ci has zero measure in the space of billiard trajectories; in this case the particle hit a
singular point of B and the motion is not defined since then.)
If x > pi(y), the particle does not hit B. Indeed, by (4), the vertical straight line
(x, y, ∗) does not intersect Byz, and by (6), it does not intersect Byx. The other sets
Bxy, Bxz, Bzx, Bzy comprising B belong to the galleries Gx and Gz, and therefore do not
intersect this straight line.
Assume that x < pi(y) and consider an auxiliary trajectory with the same initial
data making reflections from ˜Ayz. Like in the case (i), there are 4 reflections; the first
three segments of the trajectory (between the point of entering Q and the 1st reflection
point; between the 1stand the 2nd reflection points; between the 2nd and the 3rd reflection
points) will be referred to as 01-segment, 12-segment, and 23-segment, respectively. The trajectory is symmetric with respect to the xy-plane, and the final velocity, (0, 0, −1), coincides with the initial one.
Repeating the argument of (i), one concludes that the 1stand 2nd reflection points and
the 12-segment joining them belong to Πz, and by symmetry, the 3rd and 4th reflection
points and the segment joining them also belong to Πz. Therefore all the reflection points
belong to Byz. It remains to check that the auxiliary trajectory does not intersect B \ Byz,
and therefore, is a true trajectory in the complement of B.
Recall that the sets Bxy, Bxz, Bzx, Bzybelong to Gx∪Gz, and therefore do not intersect
the trajectory. It remains to check the set Byx. Since the 01- and the 12-segments belong
to Πz, they do not have points in common with Byx. The same is true for the last two
segments symmetric to them. It remains to check the 23-segment.
Let (x(1), y(1), z(1)) and (x(2), y(2), z(2)) be the points of first and second reflection.
Notice that the 23-segment belongs to the straight line (x(2), y(2), ∗). One has
ci+2 < y(2) < ci+1, (7)
x(2) = x(1) = x, 0 < y(2) < y(1) = y, p
i(y(1)) < qi+1(y(2)), and therefore,
x(2) < q
i+1(y(2)). (8)
By (7), (8), and (6), the straight line (x(2), y(2), ∗) does not intersect B
yx. Theorem is
proved.
Remark 4. We do not know if it is possible to generalize our construction to 3 non-orthogonal directions. The direct generalization does not work even in the case where two non-orthogonal directions lie in the xy-plane and the third one coincides with the
(a) (b) (c)
Figure 13: Non-overlapping bodies invisible in different directions: (a) along the z-axis; (b)-(c): along the y- and x-axes respectively.
z-direction. Indeed, a particle falling vertically down and hitting a mirror in the gallery Gy, will then move in a direction orthogonal to the y-axis (and not parallel to the x-axis,
which would be desirable), and therefore may fail to hit the opposite mirror in that gallery.
5
Summary
We have shown that there exist bodies invisible in 2 directions in two-dimensional case and in 3 directions in the three-dimensional case. It was not known earlier whether such bodies exist. We believe that our construction can be more or less directly generalized to n directions of invisibility for n-dimensional bodies, n > 3.
There are, however, many open questions. Can we construct a body invisible in n di-rections in n-dimensional space without using any fractal constructions? Are there bodies invisible in more than n directions in n-dimensional space? What is the maximal number of directions of invisibility? How to introduce an adequate ”measure of invisibility” for a body observed in all directions and find the ”most invisible” body?
There is an intriguing observation related to the existing constructions. There exist connected (and even homeomorphic to the ball) bodies invisible in 1 direction [1]. The body invisible in 2 directions found in [10] is disconnected. The body invisible in 3 directions has infinite number of connected components. We wonder if the increased complexity of the shape is the cost one should pay for the increased number of directions, or it is just an artefact of the particular constructions.
These problems are easy to understand, and the existing results can be explained by using only basic school math. However, there are no tools or techniques for constructing invisible bodies, and this makes the subject even more exciting.
Figure 14: A body invisible in 3 directions
Acknowledgements
This work was partly supported by the Center for Research and Development in Math-ematics and Applications (CIDMA) from the ”Funda¸c˜ao para a Ciˆencia e a Tecnologia” (FCT), cofinanced by the European Community Fund FEDER/POCTI, and by the FCT research project PTDC/MAT/113470/2009.
References
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